JEE Advanced Integrals — previous year questions

55 questions from 2025, 2024, 2023, 2022, 2021, 2020, 2019, 2018, 2017, 2016, 2015, 2014, 2013, 2012, 2011, 2010, 2009, 2008, 2007.

• For all x > 0, let y₁(x), y₂(x), y₃(x) be the functions satisfying dy₁/dx - (sin x)² y₁ = 0, y₁(1) = 5,…2025JEE Advanced
• If α = ∫_(1/2)² (tan⁻¹ x)/(2x² - 3x + 2) dx, then the value of √7tan(2α√7/π) is ________. (Here, the inverse…2025JEE Advanced
• Let α and β be the real numbers such that lim_(x → 0) 1/x³ ( α/2 ∫₀ˣ 1/(1-t²) dt + β x cos x ) = 2. Then the…2025JEE Advanced
• The value of the definite integral ∫₀²1/(3ˣ+3) dx is2024JEE Advanced
• Let the function f: [1, ∞) → R be defined by f(t) = (-1)ⁿ⁺¹2, if t = 2n-1, n ∈ N, ((2n+1-t))/2 f(2n-1) +…2024JEE Advanced
• Let f : (0,1) → R be the function defined as f(x) = √n if x ∈ [1/(n+1), 1/n) where n ∈ N. Let g : (0,1) → R…2023JEE Advanced
• For x ∈ R, let tan⁻¹(x) ∈ (-π/2, π/2). Then the minimum value of the function f : R → R defined by f(x) =…2023JEE Advanced
• For positive integer n, define f(n) = n + (16+5n-3n²)/(4n+3n²) + (32+n-3n²)/(8n+3n²) + (48-3n-3n²)/(12n+3n²)…2022JEE Advanced
• The greatest integer less than or equal to ∫₁²log₂(x³+1) dx+∫₁^(log₂9)(2ˣ-1)^(1/3) dx is ________.2022JEE Advanced
• Consider the equation ∫₁ᵉ ((logₑ x)^(1/2))/(x(a-(logₑ x)^(3/2))²) dx = 1, a ∈ (-∞,0)∪(1,∞).2022JEE Advanced
• For any real number x, let [x] denote the largest integer less than or equal to x. If I =…2021JEE Advanced
• Let f:[-π/2, π/2] → R be a continuous function such that f(0) = 1 and ∫₀^(π/3) f(t)dt = 0. Then which of the…2021JEE Advanced
• Let b be a nonzero real number. Suppose f : R → R is a differentiable function such that f(0) = 1. If the…2020JEE Advanced
• Let f : R → R and g : R → R be functions satisfying f(x+y) = f(x) + f(y) + f(x)f(y) and f(x) = xg(x) for all…2020JEE Advanced
• Let f:R → R be a differentiable function such that its derivative f' is continuous and f(π) = -6. If F:[0, π]…2020JEE Advanced
• Which of the following inequalities is/are TRUE?2020JEE Advanced
• If I=2/π∫_(-π/4)^(π/4)dx/((1+e^(sin x))(2-cos 2x)) , then 27 I² equals ___,2019JEE Advanced
• The value of the integral ∫₀^(π/2) 3√(cosθ)/((√(cosθ)+√(sinθ))⁵) dθ equals ___2019JEE Advanced
• Let f: R → R be given by f(x)=(x-1)(x-2)(x-5). Define F(x)=∫₀ˣ f(t) dt, x>0. Then which of the following…2019JEE Advanced
• For a ∈ R, |a| > 1, let limlimits_(x → ∞) ( (1 + ³√2 + … + ³√n)/(n^(7/3)( 1/((an+1)²) + 1/((an+2)²) + … +…2019JEE Advanced
• The value of the integral ∫₀^(1/2) (1+√3)/(((x+1)²(1-x)⁶)^(1/4)) dx is ______.2018JEE Advanced
• Let f:R→R be a differentiable function such that f(0)=0, f(π/2)=3 and f'(0)=1. If…2018JEE Advanced
• For each positive integer n, let yₙ = [1/n(n+1)(n+2)…(n+n)]^(1/n) For x ∈ R, let [x] be the greatest integer…2018JEE Advanced
• If I=∑ₖ₌₁⁹⁸∫ₖᵏ⁺¹(k+1)/(x(x+1)) dx, then2017JEE Advanced
• Let f : R → (0,1) be a continuous function. Then, which of the following function(s) has(have) the value zero…2017JEE Advanced
• The value of ∫_(-π/2)^(π/2) (x²cos x)/(1+eˣ) dx is equal to2016JEE Advanced
• The total number of distinct x ∈ [0, 1] for which ∫₀ˣ t²/(1+t⁴) dt = 2x - 1 is2016JEE Advanced
• Let f(x)=lim_(n→∞)((nⁿ(x+n)(x+n/2)⋯(x+n/n))/(n!(x²+n²)(x²+n²/4)⋯(x²+n²/n²)))^(x/n), for all x>0. Then2016JEE Advanced
• If α = ∫₀¹ (e^(9x+3tan⁻¹x)) ((12+9x²)/(1+x²)) dx where tan⁻¹x takes only principal values, then the value of…2015JEE Advanced
• Let f: R → R be a function defined by f(x) = [x], x ≤ 2 0, x > 2 where [x] is the greatest integer less than…2015JEE Advanced
• Let f'(x) = 192x³/(2+sin⁴ π x) for all x ∈ R with f(1/2) = 0. If m ≤ ∫_(1/2)¹ f(x) dx ≤ M, then the possible…2015JEE Advanced
• Let f(x) = 7tan⁸x + 7tan⁶x - 3tan⁴x - 3tan²x for all x ∈ (-π/2, π/2). Then the correct expression(s) is(are)2015JEE Advanced
• The option(s) with the values of a and L that satisfy the following equation is(are) (∫₀^(4π) eᵗ(sin⁶ at +…2015JEE Advanced
• Let f: R → R be a continuous odd function, which vanishes exactly at one point and f(1) = 1/2. Suppose that…2015JEE Advanced
• The following integral ∫_(π/4)^(π/2) (2cosec x)¹⁷ dx is equal to2014JEE Advanced
• Let f : [0, 2] → R be a function which is continuous on [0, 2] and is differentiable on (0, 2) with f(0) = 1.…2014JEE Advanced
• The value of ∫₀¹ 4x³{d²/dx²(1-x²)⁵}dx is2014JEE Advanced
• Let f: (0, ∞) → R be given by f(x) = ∫_(1/x)ˣ e^(-(t + 1/t)) dt/t. Then2014JEE Advanced
• Let f : [a, b] → [1, ∞) be a continuous function and let g : R → R be defined as g(x) = 0, if x < a, ∫ₐˣ f(t)…2014JEE Advanced
• | List-I | List-II | | --- | --- | | (P) The number of polynomials f(x) with non-negative integer…2014JEE Advanced
• Let f:[1/2,1]→ R (the set of all real number) be a positive, non-constant and differentiable function such…2013JEE Advanced
• The integral ∫ sec²x/((sec x+tan x)^(9/2)) dx equals (for some arbitrary constant K)2012JEE Advanced
• The value of the integral ∫_(-π/2)^(π/2) ( x² + ln(π+x)/(π-x) )cos x dx is2012JEE Advanced
• The value of ∫_(√(ln 2))^(√(ln 3)) (x sin x²)/(sin x² + sin(ln 6 - x²)) dx is2011JEE Advanced
• Let f be a real-valued function defined on the interval (-1,1) such that e⁻ˣf(x)=2+∫₀ˣ√(t⁴+1) dt, for all…2010JEE Advanced
• For any real number x, let [x] denote the largest integer less than or equal to x. Let f be a real valued…2010JEE Advanced
• Let f be a real-valued function defined on the interval (0, ∞) by f(x)=ln x+∫₀ˣ√(1+sin t) dt. Then which of…2010JEE Advanced
• The value(s) of ∫₀¹(x⁴(1-x)⁴)/(1+x²) dx is (are)2010JEE Advanced
• The value of lim_(x→ 0) 1/x³ ∫₀ˣ (t ln(1+t))/(t⁴+4) dt is2010JEE Advanced
• Let f be a non-negative function defined on the interval [0, 1]. If ∫₀ˣ √(1-(f'(t))²) dt = ∫₀ˣ f(t) dt, 0 ≤ x…2009JEE Advanced
• Let f:R→R be a continuous function which satisfies f(x)=∫₀ˣf(t) dt. Then the value of f(ln 5) is2009JEE Advanced
• If Iₙ = ∫_(-π)^π (sin nx)/((1+πˣ)sin x) dx, n = 0, 1, 2, …, then2009JEE Advanced
• Let I=∫ eˣ/(e⁴ˣ+e²ˣ+1) dx, J=∫ e⁻ˣ/(e⁻⁴ˣ+e⁻²ˣ+1) dx. Then, for an arbitrary constant C, the value of J-I…2008JEE Advanced
• Let f(x) be a non-constant twice differentiable function defined on (-∞, ∞) such that f(x)=f(1-x) and…2008JEE Advanced
• Let F(x) be an indefinite integral of sin² x. \ STATEMENT 1 : The function F(x) satisfies F(x+π) = F(x) for…2007JEE Advanced

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